Category theoretic proof of Stone's representation theorem

Question

Can Stone's representation theorem about Boolean algebras that every Boolean algebra $B$ is isomorphic to the algebra of clopen subsets of its Stone space $S(B)$, be proven categorically using Yoneda lemma ?

Answer 1

Here is a category-theoretic proof of a version of Stone's theorem; the idea is that we will deduce Stone's theorem for all Boolean algebras from Stone's theorem for finite Boolean algebras. So, first prove in any way you like that the category of finite Boolean algebras is equivalent to the opposite of the category of finite sets. Next, the category of Boolean algebras is, for general categorical reasons, the ind-category of the category of finite Boolean algebras, so its opposite is

$$(\text{Ind}(\text{FinBool}))^{op} \cong \text{Pro}(\text{FinBool}^{op}) \cong \text{Pro}(\text{FinSet})$$

so the opposite of the category of Boolean algebras is the category of profinite sets. There's some additional work to do to match this up with the usual statement of Stone's theorem.